A classification of $G_{\delta\sigma}$ zero-dimensional spaces?

Among separable metrizable spaces:

Cantor set is the unique compact zero-dimensional space without isolated points.

$$\mathbb Q$$ is the unique countable space without isolated points

$$\mathbb R \setminus \mathbb Q$$ is the unique zero-dimensional, $$G_\delta$$-space with no compact neighborhood.

$$\mathbb Q ^\omega$$ is the unique zero- dimensional, first category $$F_{\sigma\delta}$$-space with the property that no nonempty clopen subset is a $$G_{\delta\sigma}$$-space.

Question. Is there a simple classification of zero-dimensional $$G_{\delta\sigma}$$-spaces which have no compact neighborhoods? The simplest examples would be $$\mathbb Q$$, $$\mathbb R\setminus \mathbb Q$$, and $$\mathbb Q\times (\mathbb R\setminus \mathbb Q)$$. Are there many others?

Is there a nice characterization of $$\mathbb Q\times (\mathbb R\setminus \mathbb Q)$$?

This paper by Van Mill from 1981 gives a characterisation of $$\Bbb Q \times \Bbb P$$ (where $$\Bbb P$$ is a common notation for the irrationals) in Thm 5.3:
If $$X$$ is separable metrisable and zero-dimensional, $$\sigma$$-complete and nowhere complete and nowhere $$\sigma$$-compact then $$X \simeq \Bbb Q \times \Bbb P$$.
Where by complete I mean topologically complete (i.e. in this context: completely metrisable) and a nowhere-$$P$$ space is one where no non-empty open subset has property $$P$$, so $$\Bbb P$$ and $$\Bbb Q$$ are nowhere locally compact, e.g.)
I think $$\Bbb Q \times C$$ (with $$C$$ the Cantor set) is another example for your list.
A lot of information can be found in van Engelen's PhD-thesis from 1985: homogeneous zero-dimensional absolute Borel sets, where he shows there are are $$\omega_1$$ many homeomorphism types of subsets of $$C$$ that are both $$F_{\sigma \delta}$$ and $$G_{\delta\sigma}$$, also separately written up here. In his thesis he also gave the first characterisation of $$\Bbb Q^\omega$$ (now relegated to the appendix of it). Look up Van Engelen's work from around that time for related results.